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mathematics
calculus with applications

Calculus With Applications 10th Edition Margaret L Lial, Raymond N Greenwell, Nathan P Ritchey - Solutions

Find dy/dx.x + 2y/x – 3y = y1/2
Find dy/dx.ln (xy + 1) = 2xy3 + 4
Find dy/dx.ln (x + y) = 1 + x2 + y3
Find dy/dt . y 1+ √x 1-√x dx dt -4, x = 4
Find dy/dt.y = 8x3 – 7x2; dx/dt = 4, x = 2
Find dy/dt . x² + 5y x - 2y 2; dx dt = 1, x = 2, y = 0
Find dy/dt.y = 9 – 4x/3 + 2x; dx/dt = –1, x = –3
Find dy/dt. y 1 et² + 1' dx dt 3, x = 1
Find dy/dt.y = xe3x; dx/dt = –2, x = 1
Evaluate dy.y = 3x – 7/2x + 1; x = 2, Δx = 0.003
Evaluate dy.y = 8 – x2 + x3; x = –1, Δx = 0.02
Integration by substitution is related to what differentiation method? What type of integrand suggests using integration by substitution?
In your own words, describe what is meant by an integrand.
Use substitution to find each indefinite integral.∫4(2x + 3)4 dx
Find the following.∫6 dk
Find the following.∫9 dy
A reasonable assumption is that over time scarcity might drive up the price of oil and thus reduce consumption. Comment on the fact that the rate of oil consumption actually increased in 2002, connecting current events and economic forecasts to the short-term possibility of a reduction in
Find the following.∫(2z + 3) dz
Develop a spreadsheet that shows the time to exhaustion for various values of k.
Find the following.∫(3x – 5) dx
Go to the website WolframAlpha.com and enter “integrate.” Follow the instructions to find the time to exhaustion for various values of k. Discuss how the solution compares with the solutions provided by a graphing calculator and by Microsoft Excel.
Find the following.∫(6t2 – 8t + 7) dt
Find the following.∫(5x2 – 6x + 3) dx
Find the following.∫(4z3 + 3z2 + 2z – 6)dz
Find the following.∫(16y3 + 9y2 – 6y + 3)dy
Find the following.∫(5√z + √2) dz
Find the following.∫(t1/4 + π1/4)dt
Find the following.∫5x(x2 – 8)dx
Find the following.∫x2(x4 + 4x + 3)dx
Find the following.∫(4√v – 3v3/2)dv
A sales manager presented the following results at a sales meeting.Find the total sales over the given period as follows.a. Plot these points. Connect the points with line segments.b. Use the trapezoidal rule to find the area bounded by the broken line of part a, the x-axis, the line x =1 , and the
Find the following.∫(15x√x + 2√x) dx
Find the following.∫(10u3/2 – 14u5/2) du
In Exercises 19–40, find each indefinite integral.∫(2x + 3) dx
Find the following.(56t5/2 + 18t7/2)dt
In Exercises 19–40, find each indefinite integral.∫(5x – 1) dx
Find the following.∫(7/z2) dz
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = ln(x2 + 1)
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x2 log |x|
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = –x/1 – x2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = √x2 + 4
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = √2x2 + 9
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 32x3/4
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = –6x1/3
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 5e–x2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 0.5ex2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = ln x/4x
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = ln x + 1/x
Find the open intervals where f is increasing or decreasing.f(x) = 8xe–4x
Find the locations and values of all relative maxima and minima.f(x) = xex/x – 1
Find the locations and values of all relative maxima and minima.f(x) = ln (3x)/2x2
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = 2e–x2
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x8/3 – 4x5/3
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x7/3 + 56x4/3
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = x4 + 2x2
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = 6x3 – x4
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = x2 + 4/x
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = x + 8/x
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = –4x/1 + 2x
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = xe2x
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = x2e2x
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = 4x1/3 + x4/3
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave down, intercepts where possible, and asymptotes where applicable.f(x) = 5x2/3 + x5/3
In 2008, the price of gasoline in the United States spiked and then dropped. The average monthly price (in cents per gallon) of unleaded regular gasoline for 2008 can be approximated by the functionfor 0 < t < 12, where t is in months and t = 1 corresponds to January 2008.a. Determine the
The accompanying figure shows the product life cycle graph, with typical products marked on it. It illustrates the fact that a new product is often purchased at a faster and faster rate as people become familiar with it. In time, saturation is reached and the purchase rate stays constant until the
Find any critical numbers for f in Exercises 57-64 and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f″(c) = 0 or f″(c) does not exist for a critical number c, then the second derivative test gives no information. In
Assume that the number of bacteria R(t) (in millions) present in a certain culture at time t (in hours) is given bya. At what time before 8 hours will the population be maximized?b. Find the maximum population. R(t) = ²(t-18) + 96t + 1000.
Under the scenario that the fertility rate in the European Union (EU) remains at 1.8 until 2020, when it rises to replacement level, the predicted population (in millions) of the 15 member countries of the EU can be approximated over the next century bywhere t is the number of years since 2000.a.
The percent of concentration of a certain drug in the bloodstream x hours after the drug is administered is given byFor example, after 1 hour the concentration is given bya. Find the time at which concentration is a maximum.b. Find the maximum concentration. K(x) 3x x² + 4'
The next two exercises are a continuation of exercises first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of each logistic function. This is the point at which the growth rate begins to decline.The growth function for a population of beetles
The next two exercises are a continuation of exercises first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of each logistic function. This is the point at which the growth rate begins to decline.The population of a bed of clams is described by
An autocatalytic chemical reaction is one in which the product being formed causes the rate of formation to increase. The rate of a certain autocatalytic reaction is given bywhere x is the quantity of the product present and 100 represents the quantity of chemical present initially. For what value
Roger Clemens, ace pitcher for many major league teams, including the Boston Red Sox, is standing on top of the 37-ft-high “Green Monster” left-field wall in Boston’s Fenway Park, to which he has returned for a visit. We have asked him to fire his famous 95 mph (140 ft per second) fastball
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.ln 0.98
The concentration of a certain drug in the bloodstream x hours after being administered is approximatelyUse the differential to approximate the changes in concentration for the following changes in x.a. 1 to 1.5 b. 2 to 2.25 C(x) 5x 9 + x²*
Every year, Corinna Paolucci sells 30,000 cases of her Famous Spaghetti Sauce. It costs her $1 per year in electricity to store a case, plus she must pay annual warehouse fees of $2 per case for the maximum number of cases she will store. If it costs her $750 to set up a production run, plus $8 per
The population of bacteria (in millions) in a certain culture x hours after an experimental nutrient is introduced into the culture isUse the differential to approximate the changes in population for the following changes in x.a. 2 to 2.5 b. 3 to 3.25 P(x) = 25x 8 + x²*
A man 6 ft tall is walking away from a lamp post at the rate of 50 ft per minute. When the man is 8 ft from the lamp post, his shadow is 10 ft long. Find the rate at which the length of the shadow is increasing when he is 25 ft from the lamp post. 6 ft
The short-term demand for crude oil in the United States in 2008 can be approximated by q = f(p) = 2,431,129p–0.06, where p represents the price of crude oil in dollars per barrel and q represents the per capita consumption of crude oil. Calculate and interpret the elasticity of demand when the
In 2008, the Valve Corporation, a software entertainment company, ran a holiday sale on its popular Steam software program. Using data collected from the sale, it is possible to estimate the demand corresponding to various discounts in the price of the software. Assuming that the original price was
Let Find each derivative.a. du/dv b. dv/duc. Based on your answers to parts a and b, what do you notice about the relationship between du/dv and dv/du? √u + √2v + 1 = 5.
Find the absolute maximum and minimum of f(x) = 2x – 3x2/3(a) On the interval [–1, 0.5];(b) On the interval [0.5, 2]
The number of bank robberies in the United States for the years 2000–2009 is given in the following figure. Consider the closed interval [2000, 2009].a. Give all relative maxima and minima and when they occur on the interval.b. Give the absolute maxima and minima and when they occur on the
Let eu2–v – v = 1. Find each derivative.a. du/dvb. dv/duc. Based on your answers to parts a and b, what do you notice about the relationship between du/dv and dv/du?
The number of bank burglaries (entry into or theft from a bank during nonbusiness hours) in the United States for the years 2000–2009 is given in the figure. Consider the closed interval [2000, 2009].a. Give all relative maxima and minima and when they occur on the interval.b. Give the
Answer the questions in Exercise 40 for two investments with respective rates of profitability P1'(t) = 130 + t2 hundred dollars per year and P2'(t) = 306 + 5t hundred dollars per year.Data from Exercises 40Suppose that t years from now, one investment plan will be generating profit at the rate of
Use the method of Lagrange multipliers to prove that of all isosceles triangles with a given perimeter, the equilateral triangle has the largest area.
Beverly estimates that when she is corresponding on a regular basis with x close friends and is working on y interesting projects, her total satisfaction is measured by the utility function U(x, y) = (x + 1)(y + 2). What is Beverly’s level of satisfaction if she is currently communicating with 25
A car traveling at 67 ft/sec decelerates at the constant rate of 23 ft/sec2 when the brakes are applied.a. Find the velocity v(t) of the car t seconds after the brakes are applied. Then find its distance s(t) from the point where the brakes are applied.b. Use the graphing utility of your calculator
What is ƒbx dx for base (b > 0, b ≠ 1)?
Consider the functiona. Using a graphing calculator, try to find any local minima, or tell why finding a local minimum is difficult for this function.b. Find any local minima using the techniques of calculus.c. Based on your results in parts a and b, describe circumstances under which relative
On August 8, 2007, the power used in New York state (in thousands of megawatts) could be approximated by the functionwhere t is the number of hours since midnight, for 0 ≤ t ≤ 24. Find any relative extrema for power usage, as well as when they occurred. P(t) = -0.005846t³ + 0.16142 0.4910t+
The demand equation for telephones at one store iswhere p is the price (in dollars) and q is the quantity of telephones sold per week. Find the values of q and p that maximize revenue. P = D(q) = 200e-0.1g,
The annual unemployment rates of the U.S. civilian non institutional population for 1990–2009 are shown in the graph. Identify the years where relative extrema occur, and estimate the unemployment rate at each of these years. Rate (%) 10 9 8 7 654321 2 14 1990 92 94 96 98 2000 '02 04 06 08 Year
a. Verify thatThis expression is called the relative rate of change. It expresses the rate of change of f relative to the size of f. Stephen B. Maurer denotes this expression by f̂ and notes that economists commonly work with relative rates of change.b. Verify thatInterpret this equation in terms
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = x2–x2
On many calculators, graphs of rational functions produce lines at vertical asymptotes. For example, graphing y = (x – 1)/(x + 1) on the window [–4.9, 4.9] by [–4.9, 4.9] produces such a line at x = –1 on the TI-84 Plus and TI-89. But with the window [–4.7, 4.7] by [–4.7, 4.7] on
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.y = 1.1 – 0.3x – 0.3x2
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = 2/3 x3 – x2 – 24 – x
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = 2/3 x3 – x2 – 4x + 2

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